Background Reservoir Computing Liquid State Machines Current and Future Research Summary Reservoir Computing with Emphasis on Liquid State Machines Alex Klibisz University of Tennessee aklibisz@gmail.com November 28, 2016
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Context and Motivation • Traditional ANNs are useful for non-linear problems, but struggle with temporal problems. • Recurrent Neural Networks show promise for temporal problems, but the models are very complex and difficult, expensive to train. • Reservoir computing provides a model of neural network/microcircuit for solving temporal problems with much simpler training.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Artificial Neural Networks • Useful for learning non-linear f ( x i ) = y i . • Input layers takes vectorized input. • Hidden layers transform the input. • Output layer indicates something meaningful (e.g. binary class, distribution over classes). 1 • Trained by feeding in many examples to minimize some objective function. 1 source: wikimedia
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Feed-forward Neural Networks • Information passed in one direction from input to output. • Each neuron has a weight w for each of its inputs and a single bias b . • Weights, bias, input used to compute the output: 1 output = j w j x j − b ) . 1+exp( − � • Outputs evaluated by objective function (e.g. classification accuracy). • Backpropagation algorithm adjusts w and b to minimize the objective function.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Recurrent Neural Networks 2 Figure: The network state and resulting output change with time. • Some data have temporal dependencies across inputs (e.g. time series, video, text, speech, movement). • FFNN assume inputs are independent and fail to capture this. • Recurrent neural nets capture temporal dependencies by: 1. Allowing cyclic connections in the hidden layer. 2. Preserving internal state between inputs. • Training is expensive; backpropagation-through-time is used to unroll all cycles and adjust neuron parameters. 2 http://colah.github.io/posts/2015-08-Understanding-LSTMs/
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Continuous Activation vs. Spiking Neurons How does a neuron produce its output? • Continuous activation neurons 1. Compute an activation function using inputs, weights, bias. 2. Pass the result to all connected neurons. • Spiking neurons 1. Accumulate and store inputs. 2. Only pass the results if a threshold is exceeded. • Advantages • Proven that spiking neurons can compute any function computed by sigmoidal neurons with fewer neurons (Maass, 1997).
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Conceptual Introduction Figure: Reservoir Computing: construct a reservoir of random recurrent neurons and train a single readout layer.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary History Random networks with a trained readout layer • Frank Rosenblatt, 1962; Geoffrey Hinton, 1981; Buonamano, Merzenich, 1995 Echo-State Networks • Herbert Jaeger, 2001 Liquid State Machines • Wolfgang Maass, 2002 Backpropagation Decorrelation 3 • Jochen Steil, 2004 Unifying as Reservoir Computing • Verstraeten, 2007 3 Applying concepts from RC to train RNNs
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Successful Applications Broad Topics • Robotics controls, object tracking, motion prediction, event detection, pattern classification, signal processing, noise modeling, time series prediction Specific Examples • Venayagamoorthy, 2007 used an ESN as a wide-area power system controller with on-line learning. • Jaeger, 2004 improved noisy time series prediction accuracy 2400x over previous techniques. • Salehi, 2016 simulated a nanophotonic reservoir computing system with 100% speech recognition on TIDIGITS dataset.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Liquid State Machines vs. Echo State Networks Primary difference: neuron implementation • ESN: neurons do not hold charge, state is maintained using recurrent loops. • LSM: neurons can hold charge and maintain internal state. • LSM formulation is general enough to encompass ESNs.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary LSM Formal Definition 4 5 • A filter maps between two functions of time u ( · ) �→ y ( · ). • A Liquid State Machine M defined M = � L M , f M � . • Filter L M and readout function f M . • State at time t defined x M ( t ) = ( L M u )( t ). • Read: filter L M applied to input function u ( · ) at time t • Output at time t defined y ( t ) = ( Mu )( t ) = f M ( x M ( t )) • Read: the readout function f applied to the current state x M ( t ) 4 Maass, 2002 5 Joshi, Maass 2004
Background Reservoir Computing Liquid State Machines Current and Future Research Summary LSM Requirements Th. 3.1 Maass 2004 Filters in L M satisfy the point-wise separation property Class CB of filters has the PWSP with regard to input functions from U n if for any two functions u ( · ) , v ( · ) ∈ U n with u ( s ) � = v ( s ) for some s ≤ 0, there exists some filter B ∈ CB such that ( Bu )(0) � = ( Bv )(0). Intuition: there exists a filter that can distinguish two input functions from one another at the same time step. Readout f M satisfies the universal approximation property Class CF of functions has the UAP if for any m ∈ N , any set X ⊆ R m , any continuous function h : X �→ R , and any given ρ > 0, there exists some f in CF such that | h ( x ) − f ( x ) | ≤ ρ for all x ∈ X . Intuition: any continuous function on a compact domain can be uniformly approximated by functions from CF.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Examples of Filters and Readout Functions Filters Satisfying Pointwise Separation Property • Linear filters with impulse responses h ( t ) = e − at , a > 0 • All delay filters u ( · ) �→ u t 0 ( · ) • Leaky Integrate and Fire neurons • Threshold logic gates Readout functions satisfying Universal Approximation Property • Simple linear regression • Simple perceptrons • Support vector machines
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Building, Training LSMs In General • Take inspiration from known characteristics of the brain. • Perform search/optimization to find a configuration. Example: Simulated Robotic Arm Movement (Joshi, Maass 2004) • Inputs: x,y target, 2 angles, 2 prior torque magnitudes. • Output: 2 torque magnitudes to move the arm. • 600 neurons in a 20 x 5 x 6 grid. • Connections chosen from distribution favoring local conections. • Neuron and connection parameters (e.g. firing threshold) chosen based on knowledge of rat brains. • Readout trained to deliver torque values using linear regression.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Figure: Reservoir architecture and control loop from Joshi, Maass 2004
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Research Trends 1. Hardware implemenations: laser optics and other novel hardware to implement the reservoir. 2. Optimizing reservoirs: analytical insight and techniques to optimize a reservoirs for specific task. (Current standard is intuition and manual search.) 3. Interconnecting modular reservoirs for more complex tasks.
Background Reservoir Computing Liquid State Machines Current and Future Research Summary Summary • Randomly initialized reservoir and a simple trained readout mapping. • Neurons in the reservoir and the readout map should satisfy two properties for universal computation. • Particularly useful for tasks with temporal data/signal. • More work to be done for optimizing and hardware implementation.
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